Commutant of Multiplication Operators in Weighted Bergman Spaces on Polydisk
- PDF / 152,092 Bytes
- 15 Pages / 499 x 709 pts Page_size
- 68 Downloads / 254 Views
Czechoslovak Mathematical Journal
15 pp
Online first
COMMUTANT OF MULTIPLICATION OPERATORS IN WEIGHTED BERGMAN SPACES ON POLYDISK Ali Abkar, Qazvin Received November 5, 2018. Published online January 16, 2020.
Abstract. We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the n-dimensional complex plane. Characterization of the commutant of such operators is given. Keywords: multiplication operator; commutant of an operator; weighted Bergman space MSC 2010 : 47B38, 46E22, 30H20, 32A36
1. Introduction Let D denote the open unit disk in the complex plane. We mean by polydisk the set Dn = D × . . . × D of the n-dimentional complex space. For α > −1, we define the weighted Bergman space A2α (D) as the space of analytic functions f in D for which Z
D
|f (z)|2 dAα (z) < ∞,
where dAα (z) = π−1 (α + 1)(1 − |z|2 )α dx dy is the normalized area measure in the complex plane. It is well-known that A2α (D) equipped with the inner product hf, gi = (α + 1) DOI: 10.21136/CMJ.2020.0494-18
Z
D
f (z)g(z)(1 − |z|2 )α dA(z) 1
is a Hilbert space of analytic functions. It follows from the boundedness of point evaluation functional together with the Riesz’ representation theorem that A2α (D) is a reproducing kernel Hilbert space of analytic functions, and that every function f ∈ A2α (D) can be written as f (w) = hf, kw i = (α + 1)
Z
D
f (z)kw (z)(1 − |z|2 )α dA(z),
where kw (z) =
w ∈ D,
1 (1 − zw)α+2
is the reproducing kernel for the Hilbert space A2α (D). Now let Hol(Dn ) denote the space of holomorphic functions on the polydisk Dn . The weighted Bergman space on the polydisk Dn is defined by A2α (Dn ) = Hol(Dn ) ∩ L2 (Dn , dVα ), where dVα = dAα (z1 ) . . . dAα (zn ). In other words, a function f (z1 , . . . , zn ) ∈ Hol(Dn ) belongs to A2α (Dn ) if kf k2A2α (Dn ) =
Z
Dn
|f (z1 , . . . , zn )|2 dAα (z1 ) . . . dAα (zn ) < ∞,
where dAα (zk ) =
α+1 (1 − |zk |2 )α dxk dyk . π
2 It is well-known that {z n /γn }∞ n=0 is an orthonormal basis for Aα , where
γn = kz n kα = Then for f (z) =
∞ P
n=0
s
n! Γ(α + 2) . Γ(α + n + 2)
an z n we have kf k2α =
∞ P
n=0
γn2 |an |2 . Now, let β = (β1 , . . . , βn )
be a multi-index (each βi is a nonnegative integer); in this case we write β > 0. For z = (z1 , . . . , zn ) ∈ Dn we define z β = z1β1 . . . znβn and eβ = z β /γβ1 . . . γβn . With this notation, {eβ }β>0 is an orthonormal basis for A2α (Dn ). The reproducing kernel associated to the points (z1 , . . . , zn ) and w = (w1 , . . . , wn ) of the polydisk is given by (see [13]) Kz (w) = 2
n Y
1 = kz1 (w1 ) . . . kzn (wn ). (1 − z wj )α+2 j j=1 Online first
Given a bounded linear operator T on a Hilbert space H, we mean by the commutant of T the set of all bounded linear operators on H which commute with T . If we denote the algebra of all bounded linear operators on H by B(H), then the commutant of T which is denoted by (T )′ is by definition (T )′ = {S ∈ B(H) : ST = T S}. The operator of multiplication by z k ,
Data Loading...
